Geodesic Lévy flights on Zoll surfaces

We study the mean first capture time of isotropic Lévy flights on Zoll surfaces, namely the expected time for a geodesic Lévy process to reach a shrinking geodesic ball. While the leading-order asymptotics are universal, we prove that the first correction term encodes subtle geometric information. More precisely, it is completely determined by the local singularity type of the conjugate locus, quantified by the degree of the conjugate point. This yields a hierarchy of asymptotic regimes governed by the Lévy exponent.